Since the inequality \(\mathcal {L}(X)\le \mathcal {C}(X)\)  holds for every simplicial complex by Wegner's theorem [1]}, we are only left to verify that \(\mathcal {C}(X)\le \mathcal {L}(X)\)  whenever \(X\)  is vertex decomposable.
If \(X\)  is a simplex, there is nothing to prove. Otherwise, let \(v\)  be a shedding vertex of \(X\) . It then follows from [2]} that
\(\mathcal {L}(X)=\max \lbrace \mathcal {L}(\operatorname{del}(X;v)),\mathcal {L}(\operatorname{lk}(X;v))+1\rbrace \) . Therefore, once we apply to an induction together with Lemma REF , we conclude that
\(\mathcal {C}(X)&\le \max \lbrace \mathcal {C}(\operatorname{del}(X;v)),\mathcal {C}(\operatorname{lk}(X;v))+1\rbrace \\&\le \max \lbrace \mathcal {L}(\operatorname{del}(X;v)),\mathcal {L}(\operatorname{lk}(X;v))+1\rbrace \\&=\mathcal {L}(X).\) 
